The no-hair theorem states that a stationary black hole in Einstein-Maxwell theory is only determined by the mass, electric charge, and angular momentum [1]}, [2]}, [3]}, i.e., it is described by the KN black hole solutions. However, the hairy black holes can be found when the gravity couples to some extra field degree of freedom, such as the Yang-Mills fields [4]}, [5]}, [6]}, dilaton filed [7]}, and Skyrme hairs [8]}, [9]}. Most recently, by considering the complex scalar field without the static symmetry, Herdeiro and Radu [10]}, [11]}, [12]} presented a family of spinning scalar-hairy black hole solutions in Einstein gravity minimally coupled to a massive complex scalar field. Moreover, Hong, Suzuki, and Yamada [13]} also showed that there exists static and spherical charged scalar hair after the scalar mass term is considered. Additionally, the Astro observation [14]} implies that there exists dark matter [15]} and dark energy [16]} in real universe. It is natural to ask whether the WCCC is valid in these hairy black holes. In this paper, we would like to extend the Sorce-Wald method to prove the WCCC of static hairy black holes for the Einstein gravity coupled to some additional matter fields without using the explicit expressions of the metric and Lagrangian for matter fields.
